Unique Prime
Factorization

The
Fundamental
Theorem of Arithmetic
states that every natural number greater than
1
can be written as a product of
prime numbers
, and that up
to rearrangement of the factors, this product is
unique
. This is
called the
prime
factorization
of the number.

Example:

36
can be written as
6
×
6
, or
4
×
9
, or
3
×
12
, or
2
×
18
. But there is only one way to write it as a product where all the factors are primes:

36
=
2
×
2
×
3
×
3

This is the prime factorization of
36
, often written with exponents:

36
=
2
2
×
3
2

For a prime number such as
13
or
11
, the prime factorization is simply itself. Any
composite
number (that is, a whole number with more than two factors) has a non-trivial prime factorization.

The prime factorization of a number can be found using a
factor tree
. Start by finding two factors which, multiplied together, give the number. Keep splitting each branch of the tree into a pair of factors until all the branches terminate in prime numbers.

Here is a factor tree for
1386
. We start by noticing that
1386
is even, so
2
is a factor. Dividing by
2
, we get
1386
=
2
×
693
, and we proceed from there.